CONSIMILARITY OF COMMUTATIVE QUATERNION MATRICES

Transcription

1 Miskolc Mathematical Notes HU e-issn Vol. 6 (25), No. 2, pp DOI:.854/MMN ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES HIDYET HÜD KÖSL, MHMUT KYIǦIT, ND MURT TOSUN Received 3 November, 24 bstract. In this paper, the consimilarity of complex matrices is generalized for commutative quaternion matrices. In this regard, the coneigenvalue and coneigenvector for commutative quaternion matrices are defined. lso, the existence of solution to the some commutative quaternion matrix equations is characterized and solutions of these matrix equations are derived by means of real representations of commutative quaternion matrices. 2 Mathematics Subject lassification: 533; 58 Keywords: commutative quaternion, commutative quaternion matrix, consimilarity, coneigenvalue. INTRODUTION In the middle of 9th century, Sir William Hamilton defined the set of real quaternions which are denoted by [3] where K D fq D q q i q 2 j k Iq s 2 R;s D ;;2;3g i 2 D j 2 D k 2 D ; ij D j i D k; ik D ki D j; jk D kj D i: There are many applications of these quaternions. One of them is also about matrix theory. The study of the real quaternion matrices began in the first half of the 2th century, [3]. So, aker discussed right eigenvalues of the real quaternion matrices with a topological approach in []. On the other hand, Huang and So introduced left eigenvalues of real quaternion matrices [6]. fter that Huang discussed the consimilarity of the real quaternion matrices and obtained the Jordan canonical form of the real quaternion matrices under consimilarity [5]. Jiang and Wei studied the real quaternion matrix equation X ex D by means of real representation of the real quaternion matrices, [8]. lso, Jiang and Ling studied the problem of solution of the quaternion matrix equation ex X D via real representation of a quaternions matrix [7]. c 25 Miskolc University Press

2 966 HIDYET HÜD KÖSL, MHMUT KYIǦIT, ND MURT TOSUN fter Hamilton had discovered the real quaternions, Segre defined the set of commutative quaternions, []. ommutative quaternions are decomposable into two complex variables [2]. The set of commutative quaternions is 4-dimensional like set of quaternions. ut this set contains zero-divisor and isotropic elements. There are a lot of works associate with commutative quaternions. atoni et al. gave a brief survey on commutative quaternions [2]. They introduced functions of commutative quaternionic variables and obtained generalized auchy-riemann conditions. Geometrical introduction of commutative quaternions were considered by Severi in association with functions of two complex variable [2]. Differential properties of commutative quaternion functions were studied by Scorza-Drogoni []. lso, Kosal and Tosun considered commutative quaternion matrices. Moreover, they investigated commutative quaternion matrices using properties of complex matrices [9]. 2. LGERI PROPERTIES OF OMMUTTIVE QUTERNIONS set of commutative quaternions is denoted by [2] where H D fq D t xi yj k W t;x;y; 2 R and i;j;k Rg i 2 D k 2 D ; j 2 D ; ij D j i D k; ik D ki D j; jk D kj D i: There exist three kinds of conjugate of q D t xi yj k, q D t xi yj k; eq D t xi yj k; eq D t xi yj k and the norm is defined by kqk 4 D h q:q :eq:eq D.t y/ 2.x / 2ih.t y/ 2.x / 2i : Multiplication of any commutative quaternions q D t xi yj k and q D t x i y j k are defined in the following ways, qq D.tt xx yy /.xt tx y y /i.ty yt x x /j. t t xy yx /k : It is nearby to identify a commutative quaternion q 2 H with a real vector q 2 R 4 : Such an identification is denoted by q D t xi yj k Š q D t x y :

3 ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES 967 Then multiplication of q and q can be shown by using ordinary matrix multiplication t x y t qq D q q Š L q q D x t y y t x : x y y x t where L q is called the fundamental matrix of q. Theorem ([9]). If q and q are commutative quaternions and ; 2 are real numbers, then the following identities hold: / q D q, L q D L q 2/ L qq D L q L q 3/ L q 2 q D L q 2 L q 4/ T race L q D q q eq eq D 4t; kqk D det L q 4 : Theorem 2 ([9]). Every commutative quaternion can be represented by a 2x2 matrix with complex entries. Proof. Let q D t xi yj k 2 H; then every commutative quaternion can be uniquely expressed as q D c c 2 j where c D t xi; c 2 D y i are complex numbers. The linear map ' q W H! H is defined by ' q.p/ D qp for all p 2 H: This map is bijective and ' q./ D c c 2 j ' q.j / D c 2 c j with this transformation, the commutative quaternions can be seen as subsets of the matrix ring M 2./; the set of 2x2 matrices Then, H and N are essentially same. c c N D 2 W c c 2 c ;c 2 2 : Definition. Two commutative quaternions q and q are said to be consimilar if there exists a commutative quaternion p; kpk such that p q p D q I this is written as q c q : Theorem 3. For three commutative quaternion q; q ; q 2 2 H; the following statements hold: Reflexive: q c q: Symmetric: if q c q ; then q c q; Transitive: if q c q ; q c q2 ; then q c q 2 :

4 968 HIDYET HÜD KÖSL, MHMUT KYIǦIT, ND MURT TOSUN Proof. Reflexive q D q trivially, for q 2 H: So, consimilarity is reflexive. Symmetric: Let p q p D q : s p is nonsingular, we have p q p D p p q p p D q: So, consimilarity is symmetric. Transitive: Let p q p D q and p 2 q p 2 D q 2 : Then So, consimilarity is transitive. q 2 D p 2 p q p p 2 D.p 2 p /q.p 2 p / : Then, c is an equivalence relation on commutative quaternions. Obviously the consimilar commutative quaternions have the same norm. 3. ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES The set of all m n commutative quaternion matrices, which is denoted by H mn, with ordinary matrix addition and multiplication is a ring with unit. For D a ij mn 2 H mn ; the matrices D a ij mn ; e D fa ij mn and e D fa ij are conjugates mn of and T is transpose matrix of. Theorem 4 ([9]). For any 2 H mn and 2 H ns ; the followings are satisfied: T i. D T ; e D ; T ; e T D T ii../ T D T T ; iii. If ; 2 H nn are nonsingular, then./ D ; iv. D ; e D e e ; e D e : e Definition 2. matrix 2 H nn is said to be similar to a matrix 2 H nn if there exists a nonsingular matrix P 2 H nn such that P P D : The relation, is similar to, is denoted : is an equivalence relation on H nn. Definition 3. matrix 2 H nn is said to be consimilar a matrix 2 H nn if there exists a nonsingular matrix P 2 H nn such that P P D : The relation, is consimilar to, is denoted c : c is an equivalence relation on H nn. learly if 2 nn, then D Thus, if 2 nn is consimilar to 2 nn as complex matrices, is consimilar to as commutative quaternion matrices. Then, consimilarity relation in H nn is a natural extension of complex consimilarity in nn (for complex consimilarity see reference [4]).

5 ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES 969 Definition 4. Let 2 H nn ; 2 H: If there exists x 2 H n such that x D x then is called a coneigenvalues of and x is called a coneigenvector of associate with : The set of all coneigenvalues is defined as D 2 H W x D x; for some x : Recall that if x 2 H n.x /, and 2 H satisfying x D x, we call x an eigenvector of, while is an eigenvalue of. Theorem 5. If 2 H nn is consimilar to 2 H nn then, coneigenvalues of and are the same. Proof. Let : c Then, there exists a nonsingular matrix P 2 H nn such that D P P : Let 2 H be a coneigenvalue for the matrix ; then we can find a matrix x 2 H n such that x D x; x : Let y D P x: Then y D P P y D P P P x D P x D P x D y: Theorem 6. Let 2 H nn ; then is coneigenvalue of if and only if for any kˇk; ˇˇ is a coneigenvalue of : Proof. From x D x; we get xˇ D x ˇ ˇ ˇ : Definition 5 ([9]). Let D j 2 H nn where s 2 nn ; s D ;: The 2n 2n matrix is called the complex adjoint matrix of and denoted by. It is nearby to identify a commutative quaternion matrix 2 H nn with a complex matrix 2 2nn : y the Š symbol, we will denote D j Š D 2 2nn : Then, the multiplication of 2 H nn and 2 H nn can be represented by an ordinary matrix product Š./ : Theorem 7. Let ; 2 H nn ; the followings are satisfied: i..i n / D I 2n ; ii.. / D././; iii../ D././; iv. If is nonsingular, then..// D :

6 97 HIDYET HÜD KÖSL, MHMUT KYIǦIT, ND MURT TOSUN Theorem 8. For every 2 H nn ;./ \ D..// is the set of coneigenvalues of./: where..// D f 2 W./y D y; for some y g; Proof. Let D j 2 H nn where s 2 nn ; s D ; and 2 be a coneigenvalue of : Therefore there exists x 2 H n such that x D x: This implies. j /.x x j / D.x x j /;.x x / D x and. x x / D x : Using these equations, we can write x x D : x x Therefore, the complex coneigenvalue of the commutative quaternion matrix is equal to the coneigenvalue of the adjoint matrix./ that is./ \ D..//: 4. REL REPRESENTTION OF OMMUTTIVE QUTERNION MTRIES Let D i 2 j 3 k 2 H mn where s 2 R mn ; s D ;;2;3: We will define the linear transformation.x/ D X: We can write Then, we obtain D./ D i 2 j 3 k.i/ D i 3 j 2 k.j / D 2 3 i j k.k/ D 3 2 i j k: R4m4n : Here is called the representation of corresponding to the linear transformation.x/ D X:

7 ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES 97 It is nearby to identify a commutative quaternion matrix 2 H mn with a real matrix 2 R 4mn : y the Š symbol,we will denote D i 2 j 3 k Š D 2 2 R4mn : 3 Then, multiplication of 2 H mn and 2 H nk can be represented by an ordinary matrix product Š : Theorem 9. For commutative quaternion matrix, the following identities are satisfied: i. If 2 H mn ; then where P m D R m D. P m /. P n / D ; Q m Q n D ; R m R n D ; S m S n D I m I m I m I m I m I m I m I m ; ; Q m D S m D I I m I m I m I m I m I m I m I m (4.) ii. If ; 2 H mn then D ; iii. If 2 H mn ; 2 H nr ; in that case D. P n / D. P r /; iv. If 2 H mm ; then is nonsingular if and only if is nonsingular and. / D. P m /. P m /; v. If 2 H mm S ; e D " " e D " 2 " 2 I m I m where " D I m I m ; " 2 D I m I m I m I m vi. If 2 H mm S ;./ \ D. / ; ; ;

8 972 HIDYET HÜD KÖSL, MHMUT KYIǦIT, ND MURT TOSUN where. / D f 2 W y D y; for some y g; is the set of all eigenvalues of : Proof. y direct calculation, i., iii. and v. can be easily shown. For now we will prove ii., iv. and vi. ii. Let D i 2 j 3 k; D i 2 j 3 k 2 H mn where s ; s 2 R mn ; s D ;;2;3: Then, we have D D D iv. Suppose that 2 H mm is nonsingular, From D I 4, we have D P m D I4 and P m P m D I 4m : Then, is nonsingular and. / D P m P m : vi. Let D i 2 j 3 k 2 H mm where s 2 R mm ; s D ;;2;3 and 2 be a coneigenvalue of : Therefore, there exists a nonzero column vector x 2 H m so that x D x: Then, we can write x D x: Then complex coneigenvalue of commutative quaternion matrix is equivalent to the eigenvalue of that is./ \ D. /: 5. THE OMMUTTIVE QUTERNION MTRIX EQUTION X X D In this part, we take into consideration the commutative quaternion matrix equation X X D (5.) via the real representation, where 2 H mm ; 2 H nn and 2 H mn : We define the real representation matrix equation of the matrix equation (5.) by Y Y D : (5.2) Proposition. The equation (5.) has a solution if and only if the equation (5.2) has a solution Y D X :

9 ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES 973 Theorem. Let 2 H mm ; 2 H nn and 2 H mn : Then the equation (5.) has a solution X 2 H mn if and only if the equation (5.2) has a solution Y 2 R 4m4n, in which case, if Y is a solution to (5.2), then the matrix: X D 6.I m ii m j I m ki m / Y Qm Y Q n Rm Y R n Sm Y S n is a solution to (5.). I n ii n j I n ki n (5.3) Proof. We show that if the real matrix Y Y 2 Y 3 Y 4 Y D Y 2 Y 22 Y 23 Y 24 Y 3 Y 32 Y 33 Y 34 ; Y ab 2 R mn ; a;b D ;2;3;4 Y 4 Y 42 Y 43 Y 44 is a solution to (5.2), then the matrix represented in (5.3) is a solution to (5.). Since Qm Y Q n D Y; Rm Y R n D Y; Sm Y S n D Y; we have Q R m Y Q n Qm Y Q n D m Y R n Rm Y R n D Sm Y S n Sm Y S n D : (5.4) The last equation shows that if Y is a solution to (5.2), then Qm Y Q n; Rm Y R n and Sm Y S n are also solutions to (5.2). Thus the undermentioned real matrix: Y D 4 Y Q m Q n R m R n S m S n (5.5) is a solution to (5.2). fter calculation, we easily obtain where Y D Y Y Y 2 Y 3 Y Y Y 3 Y 2 Y 2 Y 3 Y Y Y 3 Y 2 Y Y ; (5.6) Y D 4.Y Y 22 Y 33 Y 44 /; Y D 4.Y 2 Y 2 Y 34 Y 43 /; Y 2 D 4.Y 3 Y 24 Y 3 Y 42 /; Y 3 D 4.Y 4 Y 23 Y 32 Y 4 /: (5.7)

10 974 HIDYET HÜD KÖSL, MHMUT KYIǦIT, ND MURT TOSUN From (5.7), we formulate a matrix as follows: X D Y Y i Y 2 j Y 3 k D 4.I m ii m j I m ki m /Y I m ii m j I m ki m learly the real representation of the commutative quaternion matrix X is Y : y Proposition (), X is a solution to equation (5.). Example. Solve matrix equation X i i j by using its real representation. X Real representation of given equation is X D If we solve this equation, we have Then, X D 4 X D D X I 2 ii 2 j I 2 ki 2 X 2i j j i k i j I 2 ii 2 j I 2 ki 2 : : : i D k j i j :

11 ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES 975 Let 6. PPENDIX. OMMUTTIVE QUTERNION MTRIX EQUTION X X Q D D R4m4n ; Then is called a real representation of corresponding to the linear transformation L.X/ D ex: For 2 H mn and 2 H nr the following equalities are easy to confirm. i. 2 P m 2 P m D e ; Q m Q n D ; (6.) R m R n D ; S m S n D ; (6.2) ii. D 2 P n D e 2 P r ; where Q;R;S are given by equation (4.) and 2 P m D I m I m I m I m Now, we investigate the solution of the matrix equation X ex D (6.3) by its real representation, where 2 H mm ; 2 H nn and 2 H mn : We first define the real representation matrix equation (6.3) by In the same manner, we have Y D X : : Y Y D : (6.4) Theorem. Let 2 H mm ; 2 H nn and 2 H mn : Then the equation (6.3) has a solution X 2 H mn if and only if the equation (6.4) has a solution Y 2 R 4m4n, in which case, if Y is a solution to (6.4), then the matrix: X (6.5) I m D 6.I m ii m j I m ki m / Y Qm Y Q n Rm Y R n Sm Y S ii m n j I m ki m is a solution to (6.3).

12 976 HIDYET HÜD KÖSL, MHMUT KYIǦIT, ND MURT TOSUN Proof. The proof is a routine process as was performed in Theorem. 7. PPENDIX. OMMUTTIVE QUTERNION MTRIX EQUTION X Q X D In the same manner, ' D R4m4n is called real representation of corresponding to the linear transformation L.X/ D e X: For 2 H mn and 2 H nr the following equalities are easy to confirm. i. 3 P m ' 3 P m D e ; Q m ' Q n D ' ; R m ' R n D ' ; S m ' S n D ' ; ii. ' D ' 3 P n ' D ' ' e 3 P r ; where Q;R;S are given by equation (4.) and 3 P m D I m I m I m I m Now, we define the real representation matrix equation of the matrix equation by where Y D ' X : : X e X D (7.) Y ' Y ' D ' (7.2) Theorem 2. Let 2 H mm ; 2 H nn and 2 H mn : Then the equation (7.) has a solution X 2 H mn if and only if the equation (7.2) has a solution Y 2 R 4m4n, in which case, if Y is a solution to (7.2), then the matrix: X (7.3) I m D 6.I m ii m j I m ki m / Y Qm Y Q n Rm Y R n Sm Y S ii m n j I m ki m

13 ONSIMILRITY OF OMMUTTIVE QUTERNION MTRIES 977 is a solution to (7.). Proof. The proof is a routine process as was performed in Theorem. 8. KNOWLEDGEMENTS The authors would like to thank the anonymous referees for their helpful suggestions and comments which improved significantly the presentation of the paper. REFERENES []. aker, Right eigenvalues for quaternionic matrices: a topological approach, Linear lgebra ppl., vol. 286, pp , 999, doi:.6/s (98)8-7. [2] F. atoni, R. annata, and P. Zampetti, n introduction to commutative quaternions, dv. ppl. lifford lgebras, vol. 6, pp. 28, 26, doi:.7/s6-6-2-y. [3] W. R. Hamilton, Lectures on quaternions. Dublin: Hodges and Smith, 853. [4] R.. Horn and. R. Johnson, Matrix nalysis. UK: ambridge University Press, 985. [5] L. Huang, onsimilarity of quaternion matrices and complex matrices, Linear lgebra ppl., vol. 33, pp. 2 3, 2, doi:.6/s ()266-x. [6] L. Huang and W. So, On left eigenvalues of a quaternionic matrix, Linear lgebra ppl., vol. 323, pp. 5 6, 2, doi:.6/s () [7] T. S. Jiang and S. Ling, On a solution of the quaternion matrix equation aex xb D c and its applications, dv. ppl. lifford lgebras, vol. 23, pp , 23, doi:.7/s [8] T. S. Jiang and M. S. Wei, On a solution of the quaternion matrix equation x aexb D c and its application, cta Math. Sin., vol. 2, pp , 25, doi:.7/s x. [9] H. H. Kösal and M. Tosun, ommutative quaternion matrices, dv. ppl. lifford lgebras, vol. 24, pp , 24, doi:.7/s [] G. Scorza-Dragoni, The analytic functions of a bicomplex variable (in italian), Memorie ccademia d Italia V, vol. 5, p. 597, 934. []. Segre, The real representations of complex elements and extension to bicomplex systems, Math. nn.), vol. 4, p. 43, 892, doi:.7/f [2] F. Severi, Opere matematiche, cc. Naz. Lincei, Roma, vol. 3, pp , 977. [3] L.. Wolf, Similarity of matrices in which the elements are real quaternions, ull. mer. Math. Soc., vol. 42, pp , 936, doi:.9/s uthors addresses Hidayet Hüda Kösal Sakarya University, Faculty of rts and Sciences, Department of Mathematics, Sakarya, Turkey address: hhkosalsakarya.edu.tr Mahmut kyiǧit Sakarya University, Faculty of rts and Sciences, Department of Mathematics, Sakarya, Turkey address: makyigitsakarya.edu.tr Murat Tosun Sakarya University, Faculty of rts and Sciences, Department of Mathematics, Sakarya, Turkey address: tosunsakarya.edu.tr